DECEMBER 2006
Code: D-01 / DC-01 Subject: MATHEMATICS - I
Time: 3 Hours Max. Marks: 100
NOTE: There are 9 Questions in all.
· Question 1 is compulsory and carries 20 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.
· Out of the remaining EIGHT Questions answer any FIVE Questions. Each question carries 16 marks.
· Any required data not explicitly given, may be suitably assumed and stated.
Q.1 Choose the correct or best alternative in the following: (2x10)
a. If one root of the equation 
is 
of the other root,
then K is
(A) 2 (B) 8
(C) 10 (D) 12
b. The centroid of the triangle formed by the straight lines 
is
(A) (0, 0) (B) (1, 0)
(C) (0, 1) (D) (1, 1)
c. The distance between the parallel lines 3x + 4y + 5 = 0 and 3x + 4y + 15 = 0 is
(A) 1 (B) 2
(C) 3 (D) 5
d. 
, where m 
n is equal to
(A) m (B) n
(C) m – n (D) m + n
e. If 
then 
is equal to
(A) 2 sin 4x (B) 4 sin 2x
(C) sin 4x (D) 2 sin 2x
f. 
is equal to
(A)
(B)
log 
(C) tan x + sec x (D) tan x – sec x
g. 
is equal to
(A)
(B)
(C) 1 (D) 0
h. The solution of
the differential equation 
is
(A)
(B)
(C) 
(D) 
i. The value of 
is equal to
(A)
1 (B)
(C) 
(D)
zero
j. The value of 
is
(A) 
(B)
(C) (D)
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. Show
that the coefficient of 
in the expansion of 
is double the
coefficient of in
the expansion of 
.
(8)
b. If 
and 
, where 
then prove that 
. (8)
Q.3 a. If
A + B + C =,
show that
. (8)
b. If a, b, c be the
sides opposite to the angles A, B, C of a triangle ABC, show that 
. (8)
Q.4 a. Derive the formula
for finding the area of a triangle whose vertices are 
and 
. (8)
b. Find the equation of a straight line joining the point (3, 5) to the point of intersection of the lines 4x +y = 1 and 7x – 3 y = 35. (8)
Q.5 a. Find the equation of the circle which passes through the centre of the circle
and
is concentric with the circle 
. (8)
b. Find
the focus, vertex, directrix and axis of the parabola 
. (8)
Q.6 a. Evaluate 
. (8)
b. Find , if 
. (8)
Q.7 a. Derive
the equation of the tangent and the normal to the curve 
at the point 
. (8)
b. Evaluate 
. (8)
Q.8 a. Find
the volume of the solid of revolution obtained by revolving the ellipse 
about x-axis. (8)
b. Evaluate
, for any
positive integer n. (8)
Q.9 Solve any two of the following differential equations.
(i) 
.
(ii)
.
(iii)
. (16)